Exploiting Partial Correlations in Distributionally Robust Optimization

TL;DR

This paper introduces a novel approach to distributionally robust optimization that leverages partial correlation information to create simplified, often polynomial-time solvable, reformulations for complex stochastic programming problems.

Contribution

It develops a reduced semidefinite programming formulation based on block-diagonal correlation structures, enabling efficient solutions for certain scheduling and network problems.

Findings

Polynomial-time solvable reformulations for appointment scheduling.

Efficient bounds for PERT networks and linear assignment problems.

First known polynomial-time formulation capturing partial correlation in scheduling.

Abstract

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set $\{(\bold{x}, \bold{x}\bold{x}'): \bold{x} \in \mathcal{X}\}$ where $\mathcal{X}$ is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in Project Evaluation and Review Technique (PERT) networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled.